Weyl Multipliers Research Articles

Paley inequality for the Weyl transform and its applications

Abstract In this paper, we prove several versions of the classical Paley inequality for the Weyl transform. As for some applications, we prove a version of the Hörmander’s multiplier theorem to discuss L p {L^{p}} - L q {L^{q}} boundedness of the Weyl multipliers and prove the Hardy–Littlewood inequality. We also consider the vector-valued version of the inequalities of Paley, Hausdorff–Young, and Hardy–Littlewood and their relations. Finally, we also prove Pitt’s inequality for the Weyl transform.

On wavelet polynomials and Weyl multipliers

On wavelet polynomials and Weyl multipliers

On the negative order Cesáro summability of double series with respect to block-orthonormal systems

Abstract Some statements connected with the ( C , - 1 < α < 0 , - 1 < β < 0 ) {(C,-1<\alpha<0,-1<\beta<0)} summability almost everywhere of double series with respect to block-orthonormal systems are given. In particular, using the obtained results, we can determine Weyl multipliers for establishing negative order summability of double series with respect to Δ p , q {\Delta_{p,q}} -ONS.

Homogeneous Fourier and Weyl multipliers on Sobolev spaces related to the Heisenberg group

Inspired by the work of A. Bonami and S. Poornima that a non-constant function which is homogeneous of degree 0 cannot be a Fourier multiplier on homogeneous Sobolev spaces, we establish analogous results for Fourier multipliers on the Heisenberg group Hn and Weyl multipliers on Cn acting on Sobolev Spaces.

On Weyl multipliers of the rearranged trigonometric system

We prove that the condition is necessary for an increasing sequence of numbers to be an almost everywhere unconditional convergence Weyl multiplier for the trigonometric system. This property was known long ago for Haar, Walsh, Franklin and some other classical orthogonal systems. The proof of this result is based on a new sharp logarithmic lower bound on for the majorant operator related to the rearranged trigonometric system. Bibliography: 32 titles.

Weighted norm inequalities for Weyl multipliers and fourier multipliers on the Heisenberg group

In this paper, we prove weighted norm inequalities for Weyl multipliers satisfying Mauceri’s condition. As an application, we prove certain multiplier theorems on the Heisenberg group and also show, in the context of a theorem of Weis on operator-valued Fourier multipliers, that R-boundedness of the derivative of the multiplier is not necessary for boundedness of the multiplier transform.

Sufficient conditions for convergence of multiple Fourier series with Jk-lacunary sequence of rectangular partial sums in terms of Weyl multipliers

We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in $L_2$ in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series $S_n(x;f)$ have indices $n=(n_1,\dots,n_N) \in \mathbb Z^N$, $N\ge 3$, in which $k$ $(1\leq k\leq N-2)$ components on the places $\{j_1,\dots,j_k\}=J_k \subset \{1,\dots,N\} = M$ are elements of (single) lacunary sequences (i.e., we consider the, so called, multiple Fourier series with $J_k$-lacunary sequence of partial sums). We prove that for any sample $J_k\subset M$ the Weyl multiplier for convergence of these series has the form $W(\nu)=\prod \limits_{j=1}^{N-k} \log(|\nu_{{\alpha}_j}|+2)$, where $\alpha_j\in M\setminus J_k $, $\nu=(\nu_1,\dots,\nu_N)\in{\mathbb Z}^N$. So, the "one-dimensional" Weyl multiplier -- $\log(|\cdot|+2)$ -- presents in $W(\nu)$ only on the places of "free" (nonlacunary) components of the vector $\nu$. Earlier, in the case where $N-1$ components of the index $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained in 1977 by M.Kojima in the classes $L_p$, $p>1$, and by D.K.Sanadze, Sh.V.Kheladze in Orlizc class. Note, that presence of two or more "free" components in the index $n$ (as follows from the results by Ch.Fefferman (1971)) does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $N\geq 3$ even in the class of continuous functions.

Weyl transform and Weyl multipliers associated with locally compact abelian groups

In this paper, we extend to the context of locally compact abelian groups, the notion of Weyl transform. We also define and characterize Weyl multipliers for certain function spaces. A dual space characterization is provided for the space of Weyl multipliers on \(L^p\) spaces. Finally we also study the twisted shift-invariant subspaces.

A condition for almost everywhere convergence of orthorecursive expansions

An almost everywhere convergence condition with the Weyl multiplier W111111111(n) = v n is obtained for orthorecursive expansions that converge to the expanded function in L2.

H p -Boundedness of Weyl multipliers

In this paper, we will define Hardy spaces H p ( C n ) associated with twisted convolution operators and give the atomic decomposition of H p ( C n ), where 2 n 2 n + 1 <p≤1. Then we consider the boundedness of the Weyl multiplier on H p ( C n ).MSC:42B30, 42B25, 42B35.

The boundedness of Weyl multiplier on Hardy spaces associated with twisted convolution

In this paper, we will consider the boundedness of Weyl multiplier on Hardy spaces associated with twisted convolution. In order to get our result, we need to give some characterizations of the Hardy space associated with twisted convolution. Including Lusin area integral, Littlewood–Paley g-function.

An addendum of an equivalence theorem

In a previous paper we showed that the two conditions which guarantee that a factor sequence should be a Weyl multiplier for a certain property of a given orthogonal series is equivalent to only one assumption. In the present paper it is verified that the additional condition prescribed on the Weyl multipliers is also necessary to the equivalence. Mathematics subject classification (2000): 26D15, 40A05, 42C15.

On the Rearranged Block-Orthonormal Systems

Abstract A condition is established for the sequence {Nk } when there exists a rearrangement of functions of the block-orthonormal system {φn } such that the sequence {log2 n} ({(log log n)2}) is a Weyl multiplier for the convergence (for the summability by the methods (C, α) (α &gt; 0)) almost everywhere of series with respect to the rearranged system {φνn }.

On the Absolute Summability of Series with Respect to Block-Orthonormal Systems

Abstract Theorems determining Weyl's multipliers for the summability almost everywhere by the |c, 1| method of the series with respect to block-orthonormal systems are proved. In particular, it is stated that if the sequence {ω(n)} is the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all orthogonal series, then there exists a sequence {Nk } such that {ω(n)} will be the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all series with respect to the Δ k -orthonormal systems.

ON THE ABSOLUTE SUMMABILITY OF SERIES WITH RESPECT TO BLOCK-ORTHONORMAL SYSTEMS

Theorems determining Weyl's multipliers for the summability almost everywhere by the |c, 1| method of the series with respect to block-orthonormal systems are proved. In particular, it is stated that if the sequence {ω(n)} is the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all orthogonal series, then there exists a sequence {Nk} such that {ω(n)} will be the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all series with respect to the Δk-orthonormal systems.

Weyl multipliers for invariant Sobelev spaces

A concrete characterization for theL P -multipliers (1<p<∞) for the Weyl transform is obtained. This is used to study the Weyl multipliers for Laguerre Sobolev spacesW m,p (ℂ n ). A dual space characterization is obtained for the Weyl multiplier classM W (W ,1 (ℂ n )).

-convergence almost everywhere of double Fourier series

In this paper we consider -convergence of double Fourier series (-convergence of a double number series implies that it converges in the sense of Pringsheim, over spheres, and so on.) Unextendable classes of Weyl multipliers for -convergence almost everywhere are described. In addition, close to exact sufficient conditions for -convergence almost everywhere in the spaces and are found.

Weyl multipliers, Bochner-Riesz means and special Hermite expansions

Weyl multipliers, Bochner-Riesz means and special Hermite expansions

Weyl multipliers for the unconditional convergence of series in a Franklin system

Weyl multipliers for the unconditional convergence of series in a Franklin system

REARRANGEMENTS OF FOURIER-WALSH SERIES

In this paper a method of rearranging Fourier-Walsh series is proposed that yields an essentially stronger estimate than previously known on a Weyl multiplier for unconditional convergence almost everywhere. The question of unconditional convergence almost everywhere of Fourier-Walsh series of Hω-functions is also studied. Bibliography: 8 titles.